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Chapter 7: Problem 7

Angular displacement \((\theta)\) of a flywheel varies with time as \(\theta=a t+b t^{2}+c t^{3}\) then angular acceleration is given by (a) \(a+2 b t-3 c t^{2}\) (b) \(2 b-6 t\) (c) \(a+2 b-6 t\) (d) \(2 b+6 c t\)

Short Answer

Expert verified
The angular acceleration is given by \( 2 b + 6 c t \), which corresponds to option (d).

Step by step solution

01

Find the angular velocity

First, we'll find the angular velocity \( \omega \). The angular velocity is the first derivative of the angular displacement \(\theta = a t + b t^2 + c t^3 \) with respect to time. Therefore, \( \omega = \frac{d \theta}{dt} = a + 2 b t + 3 c t^2 \)
02

Find the angular acceleration

Next, we'll find the angular acceleration \( \alpha \). The angular acceleration is the derivative of the angular velocity \( \omega \) with respect to time. Hence, \( \alpha = \frac{d \omega}{dt} = 0 + 2 b + 6 c t \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Angular Displacement
Angular displacement refers to the change in the angle as an object moves along a circular path. It describes how far the object has rotated around a specific point. This concept is particularly important in scenarios involving rotational motion, like a spinning wheel or a rotating flywheel.

Angular displacement is usually denoted by \( \theta \) and is measured in radians. It can be thought of as the angular equivalent to linear displacement, providing a way to express rotational change over time. For example, if a flywheel rotates one complete turn, its angular displacement is \( 2\pi \) radians.
  • Relation with Time: The angular displacement often varies with time, and in this exercise, it is given by a polynomial function \( \theta = a t + b t^2 + c t^3 \).
  • Real-World Applications: Understanding angular displacement helps in evaluating the efficiency and functioning of rotating machinery like engines and turbines.
Diving into Angular Velocity
Angular velocity is the rate at which an object's angular displacement changes over time. It tells us how quickly something is rotating and is denoted by \( \omega \).

To find angular velocity, we take the derivative of the angular displacement with respect to time. This gives us \( \omega = \frac{d \theta}{dt} \). For the given function \( \theta = a t + b t^2 + c t^3 \), the angular velocity is \( \omega = a + 2b t + 3c t^2 \).
  • Measurement: Angular velocity is typically measured in radians per second \( \text{rad/s} \).
  • Significance: It provides insight into the speed of rotation, crucial for sync operations in systems like watches, hard drives, and automotive engines.
Role of Derivatives in Physics
Derivatives are a key mathematical tool in physics, reflecting how a quantity changes with respect to another. They are essential in analyzing motion, be it linear or angular.

In this exercise, we used derivatives to find both angular velocity and angular acceleration:
  • The first derivative of angular displacement gives us angular velocity.
  • The second derivative gives us angular acceleration, \( \alpha = \frac{d \omega}{dt} = 2b + 6c t \).
This process helps in predicting how systems will behave under different conditions.

  • Application: Derivatives allow us to model and solve real-world problems in engineering, physics, and even economics.
  • Link to Calculus: This approach merges calculus with real-world data, enabling deeper understanding and forecasting of trends in rotational dynamics.

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Most popular questions from this chapter

The angular velocity of seconds hand of a watch will be (a) \(\frac{\pi}{60} \mathrm{rad} / \mathrm{sec}\) (b) \(\frac{\pi}{30} \mathrm{rad} / \mathrm{sec}\) (c) \(60 \pi \mathrm{rad} / \mathrm{sec}\) (d) \(30 \pi \mathrm{rad} / \mathrm{sec}\)

Let \(l\) be the moment of inertia of an uniform square plate about an axis \(A B\) that passes through its centre and is parallel to two of its sides. \(C D\) is a line in the plane of the plate that passes through the centre of the plate and makes an angle \(\theta\) with \(A B .\) The moment of inertia of the plate about the axis \(C D\) is then equal to [IIT-JEE 1998] (a) \(l\) (b) \(l \sin ^{2} \theta\) (c) \(l \cos ^{2} \theta\) (d) \(l \cos ^{2} \frac{\theta}{2}\)

A thin circular ring of mass \(M\) and radius \(R\) is rotating about its axis with a constant angular velocity \(\omega\). Four objects each of mass \(m\), are kept gently to the opposite ends of two perpendicular diameters of the ring. The angular velocity of the ring will be (a) \(\frac{M \omega}{M+4 m}\) (b) \(\frac{(M+4 m) \omega}{M}\) (c) \(\frac{(M-4 m) \omega}{M+4 m}\) (d) \(\frac{M \omega}{4 m}\)

The position of a particle is given by : \(\vec{r}=(\hat{i}+2 \hat{j}-\hat{k})\) and momentum \(\vec{P}=(\hat{i}+4 \hat{j}-2 \hat{k})\). The angular momentum is perpendicular to (a) \(X\)-axis (b) \(Y\)-axis (c) \(Z\)-axis (d) Line at equal angles to all the three axes

A ring of radrus \(0.5 m\) and mass \(10 \mathrm{~kg}\) is rotating about its diameter with an angular velocity of 20 \(\mathrm{rad} / \mathrm{s}\). Its kinetic energy is (a) \(10 J\) (b) \(100 J\) (c) \(500 J\) (d) \(250 \cdot J\)
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